Galanopoulos and Pel\(\acute{\text{a}}\) ez proved in [1]} that the operator \(\mathcal {H}_{\mu ,1}\)  is well defined in \(H^1\)  when \(\mu \)  is a Carleson measure. See [2]} for more details on Hardy spaces. Also, they obtained the following integral representation
\(\mathcal {H}_{\mu ,1}(f)(z)=\int _{[0,1)}\frac{f(t)}{1-tz}d\mu (t),\quad z\in \mathbb {D},~\mbox{for~all}~f\in H^1.\) 
